In the N-body problem, it is classical that there are conserved quantities ofcenter of mass, linear momentum, angular momentum and energy. The level sets M(c, h) ofthese conserved quantities are parameterized by the angular momentum c and the energyh, and are known as the integral manifolds. A long-standing goal has been to identify thebifurcation values, especially the bifurcation values of energy for fixed non-zero angularmomentum, and to describe the integral manifolds at the regular values. Alain Albouyidentified two categories of singular values of energy: those corresponding to bifurcations atrelative equilibria; and those corresponding to “bifurcations at infinity”, and demonstratedthat these are the only possible bifurcation values. This work completes the identification ofbifurcations for the four-body problem with equal masses, confirming that, in this setting,Albouy’s necessary conditions for bifurcation are also sufficient conditions: bifurcations ofthe integral manifolds occur at all of the singular values of energy. A recent study examinedthe bifurcations at infinity; this work evaluates the four bifurcations at relative equilibria.To establish that the topology of the integral manifolds changes at each of these values,and to describe the manifolds at the regular values of energy, the homology groups of theintegral manifolds are computed for the five energy regions on either side of the singularvalues. The homology group calculations establish that all four energy levels are indeedbifurcation values, and allows some of the global properties of the integral manifolds to beexplored.1
2020 Mathematics Subject Classification. Primary 70F07, 58F05, 57R57; Secondary 58F14 .